MathDB
Manipulating Conjugates

Source: AMC 12A #15

November 11, 2021
AMCAMC 12AMC 12 A

Problem Statement

Recall that the conjugate of the complex number w=a+biw = a + bi, where aa and bb are real numbers and i=1i = \sqrt{-1}, is the complex number w=abi\overline{w} = a - bi. For any complex number zz, let f(z)=4izf(z) = 4i\hspace{1pt}\overline{z}. The polynomial P(z)=z4+4z3+3z2+2z+1P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1 has four complex roots: z1z_1, z2z_2, z3z_3, and z4z_4. Let Q(z)=z4+Az3+Bz2+Cz+DQ(z) = z^4 + Az^3 + Bz^2 + Cz + D be the polynomial whose roots are f(z1)f(z_1), f(z2)f(z_2), f(z3)f(z_3), and f(z4)f(z_4), where the coefficients A,A, B,B, C,C, and DD are complex numbers. What is B+D?B + D?
(<spanclass=latexbold>A</span>)304(<spanclass=latexbold>B</span>)208(<spanclass=latexbold>C</span>)12i(<spanclass=latexbold>D</span>)208(<spanclass=latexbold>E</span>)304(<span class='latex-bold'>A</span>)\: {-}304\qquad(<span class='latex-bold'>B</span>) \: {-}208\qquad(<span class='latex-bold'>C</span>) \: 12i\qquad(<span class='latex-bold'>D</span>) \: 208\qquad(<span class='latex-bold'>E</span>) \: 304